5.8 Mixed-Mode

Figure 5.3: Contact handling for mixed-mode

In case of a mixed-mode simulation (5.20) is replaced by

$$\psi_{C} \psi_{C}^{} \varphi_{C}^{}$$ (5.27)

with $\varphi_{C}^{}$ being the node voltage of the circuit node connected to the device. This can be interpreted as a zero-valued voltage source connecting the circuit node to the device (see Fig. 5.3). The constitutive relation for $\varphi_{C}^{}$ follows from KCL and reads

$$\varphi_{C} \sum_{i}^{}$$ (5.28)

with I_Di being the currents of the compact models connected to the same node. In terms of the familiar MNA stamps

jx, y	$\varphi_{C}^{}$	$\psi_{C}^{}$	IC	r
$\varphi_{C}^{}$			-1	f$\varphi_{C}$k
$\psi_{C}^{}$	1	-1		f$\psi_{C}$k
IC			-1	fICk

For the same one-dimensional contact the relevant part of the system matrix reads

jx, y	$\psi_{0}^{}$	n0	$\psi_{C}^{}$	IC	$\varphi_{C}^{}$	r
$\psi_{0}^{}$	-1		1			f$\psi_{0}$k
n0		-1				fn0k
$\psi_{C}^{}$			-1		1	f$\psi_{C}$k
IC	- ${\frac{\partial \displaystyle I_{01}}{\partial \displaystyle \psi_{0}}}$	- ${\frac{\partial \displaystyle I_{01}}{\partial \displaystyle n_{0}}}$		-1		fICk
$\varphi_{C}^{}$				-1		f$\varphi_{C}$k

Eliminating $\psi_{0}^{}$, n₀, $\psi_{C}^{}$, and I_C yields the desired result

jx, y	$\varphi_{C}^{}$	r
$\varphi_{C}^{}$	- ${\frac{\partial \displaystyle I_{01}}{\partial \displaystyle \psi_{0}}}$	- f$\psi_{C}$k + fICk + fn0k . ${\frac{\partial \displaystyle I_{01}}{\partial \displaystyle n_{0}}}$ - ${\frac{\partial \displaystyle I_{01}}{\partial \displaystyle \psi_{0}}}$ . (f$\psi_{0}$k + f$\psi_{C}$k)